The research conducted under this grant will study the interface between two fields of mathematics: homotopy theory and algebraic geometry. Since the introduction of motivic homotopy theory by F. Morel and V. Voevodsky, it has been possible to use homotopy theoretic methods to study schemes (rigid geometric objects controlled by algebraic equations, important to topics as far-ranging as the theory of numbers and the laws controlling our physical universe). Meanwhile chromatic homotopy theory and topological modular forms provide tools for transferring algebro-geometric data into topology and homotopy theory. This research will work at the intersection of these fields, using algebraic geometry and homotopy theory to inform each other. By incorporating portions of this program into the thesis curriculum at Reed College, this research will enhance the educational experience of Reed's students, exposing a diverse population of undergraduates to research-level mathematics.
Ormsby proposes a four-fold program for studying chromatic and motivic homotopy theory. First, furthering joint work with J. Heller, he will explicate the fashion in which Galois theory controls a particularly tractable part of the stable motivic homotopy category. Second, continuing joint work with M. Behrens, N. Stapleton, and V. Stojanoska, he will use bivariable modular forms and the Adams spectral sequence to study cooperations in topological modular forms. Third, he plans to study the convergence properties of the motivic slice spectral sequence over infinite-cohomological dimension fields, ultimately leading to computations in stable motivic homotopy sheaves. Finally, he plans to introduce a theory of spectral presheaves with framed transfers and study their relation to foundational questions about the stable motivic homotopy category.
